Following [1]}, [2]}, we define an average length scale \(a(\tau )\)  on the flow lines of \(u\)  via \(a^{\prime }/a = H\)  where \(^{\prime }\)  denotes a derivative with respect to the proper time \(\tau \)  of the flow lines. With this definition, we have \(a^{\prime \prime }/a = H^{\prime } + H^2\)  which generalizes eq. (REF ). Note that, for FLRW spacetimes, the average length scale coincides with the scale factor.
